CC BY
33
MSC 43A85, 32D15
Analytic continuation of spherical functions for the hyperboloid of one sheet 1
© V. F. Molchanov
Derzhavin Tambov State University, Tambov, Russia
For the hyperboloid of one sheet in R3, we construct four complex hulls and continue analytically to them spherical functions of different series (continuous, holomorphic and antiholomorphic)
Keywords: hyperboloid of one sheet, complex hulls, spherical functions
In this paper we study analytic continuation of spherical functions on the hyperboloid of one sheet X in R3 on complex hulls of the hyperboloid.
The quasiregular representation of the group G = SO0(1,2) decomposes in irreducible unitary representations of the continuous series (with multiplicity 2) and the holomorphic and anti-holomorphic discrete series (with multiplicity 1), This
X
spherical functions of these series.
It was known [2], [3] that spherical functions of discrete series can be continued
X
a similar question for spherical functions of the continuous series was not studied. In this paper we construct 4 complex hulls Y + Y — H+ and H- of the hyperboloid
X
two first manifolds, each series needs its own hull. But for the spherical functions of the continuous series, the situation is more complicated and more interesting: the spherical functions of the continuous series need both manifolds H+ and H— each spherical function is half the sum of the limit values from H+ and H-,
§ 1. Complex hulls
R3
[x, y] = -Xiyi + X2V2 + X3y3, (1.1)
1 Supported by the Russian Foundation for Basic Research (RFBR): grant 09-01-00325-a, Sci. Progr. "Development of Scientific Potential of Higher School": project 1.1.2/9191, Fed. Object Progr. 14.740.11.0349 and Templan 1.5.07
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The hyperboloid of one sheet X is defined by the equation [x, x] = 1. Let XC be
X
[x, y] to the space C3 by the same formula (1.1). Then XC is the set of points x in C3 [x, x] = 1
We consider that the group G = SO0(1, 2) acts linearly on R3 from the right: x M xg. In accordance with it we write vectors in the row form. On the hyperboloid X, the group G acts transitively. The group G also acts on XC: x M xg, but of course not transitively.
XC
GG
GG
X
X
We need the group SL(2, C) and its subgroup SU(1,1). They consist respectively of matrices:
SL(2, C)
C = C U (the Riemann sphere) linear fractionally:
This action is transitive. But the subgroup SU(1,1) has three orbits on C: the open disk D : zz < 1, its exterior D' : zz > 1, and the circle S : zz =1.
Denote the group SU(1,1) by G1; then the group SL(2, C) is its complexifieation
The action x M g-lxg of the group G1 on these mat rices x is the action x M xg of the group G on vectors x E R3. It gives an homomorphism of G1 on G.
Introduce on X horospherical coordinates u,v\ (u,v) E S x S, u = v, bv
It embeds the hyperboloid X into the torus S x S, the image is the torus without {u = v}
When a point x E X is transformed by g E G, its coordinates (u,v) are transformed by a fractional linear transformation (the same for u and v): u M u ■ gl,
z M z ■ g
az + y
@z + 5
R3
x
ixl x2 + ix3
x2 — ix3 —ixl
The inverse map x M (u,v) is given bv
x3 + ix2 x3 + ix2
(1.2)
u =-------------, v =----------------
xl + i xl — i
xi
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v M v • g1; where g1 is an element in the group G1 which goes to g E G under the homomorphism G1 M G mentioned above.
Similarly we introduce horospherieal coordinates z,w on XC: a point x E XC is
( z + w 1 — zw 1 + zw \ ,
x = -------r, ---------, ^ , L3)
\i(z — w) z — w i(z — w))
the variables z, w run over the extended comp lex plane C, with the cond ition z = w.
The inverse map is given by
X3 + iX2 X3 + iX2 H
z =-----------, w =--------------. (1,4)
X1 + i x1 — i
These formulae, defined originally for X1 = ±i, are extended bv continuity to all X E XC, Namely, the points X = (i, i\, X), X = (—i, iX, X), X = 0, have horospherieal coordinates (0,i/X), (—i/X, 0), respectively, and the points X = (i, —iX,X), X = (—i, — iX, X) have horospherieal coordinates (—iX, ro) and (^o, iX), respectively. Thus, formulae (1,4) give an embedding XC M C x C, the mage is C x C without the diagonal.
There are the following relations. Let the points x and y in XC have horospherieal coordinates (z,w) and (X,^) respectively. Then
[x, y] — 1 = — 2((z — ^w — ’ (L5)
(z — w)(X — y)
[x, y] + 1 = — 2((z — /'t(w — X ■ (L6)
(z — w)(X — y)
If a point x in XC has coordinates (z,w), ^^^n the point x (it belongs
to XC too) has ^^^^^^terical coordinates (1/z, 1/w). Together with (1.5), (1/6) this gives
_ 1 = 2(1 — zz)(1- ww), (1.7)
|z — w|2
[x, x] + 1 = 2
Moreover, for the imaginary parts we have
1 — 2
1 — zw
zw
(1.8)
T zz — ww ^
Im X1 = —:------------, (1.9)
|z — w|2
X3 1 — zz • ww
1^^ = —n----------------iT" ■ L10)
x2 |1 — zw|2
In the direct product C x C let us consider 4 complex manifolds:
D x D, D' x D', D x D', D x D. (1.11)
The torus S x S is contained in the boundary of each of them. The group G1 acts on C x C diagonally: (z,w) M (z • g,w • g). It preserves all these manifolds (1.11). But
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this action is not transitive. This can be seen already when we compare dimensions: dimesion of G1 is less than dimension of each manifold (3 < 4), Further, the group G1 preserves [x,X], therefore, bv (1.8), it preserves, for instance,
j [x,x + 1 1 — zw 2
2 z — w ’
so that any G^orbit lies on the level surface J = const.
G1
(—iy,iy), 0 ^ y < 1, for D x D,
(—iy,iy), 1 < y ^ ro, for D' x D',
(—iy,iy-1), 0 ^ y < 1, for D x D',
(—iy,iy-1), 1 < y ^ ro, for D' x D.
D x D G1 D
first element of a pair in D x D to zero. We obtain a pair (0, Z), Z E D. Now we can act on this pair by the centralizer of 0, i.e the diagonal group ^ of G1. It consists
of matrices g1 with a = eia, b = 0, They act by rotations with angle 2a around zero.
So we can move Z to ir, 0 ^ r < 1. Thus, any pair in D x D can be moved to a pair
(0, ir), 0 ^ r < 1. This pair can be transferred to a pair (—iy, iy) in the lemma by
means of a matrix
1 f 1 iy \ 2y
g1 = /T—y2V —iy 1 J’ r = y+1 ■
Similarly we consider the other 3 cases. □
For all y satisfying the strong inequalities in Lemma 1.1, i. e. 0 < y < 1 or 1 < y < ro, the stabilizer of the pair indicated in the lemma is the center {±E} G1 G1
G ~ G1/{±E} and has dimension three.
For y = 0 or y = ro the stabilizer of the pairs is the subgroup K{m G1 consisting
G1
Lobachevsky plane L = G1 /K1 and have dimension two. For D x D and D' x D' these two-dimensional orbits are the diagonals {z = w}, and for D x D' and D' x D they are the manifolds {zw = 1}. Indeed, the matrix g1 carries the pairs (0, 0), (ro, ro), (0, ro), (ro, 0) to the pairs (z, z), (w, w), (z, w), (w, z) respectively, where z = b/a, w = a/b, so that zw = 1.
Let us delete these two-dimensional orbits from the manifolds (1.11) and denote the remainig manifolds by the same symbols with index 0, for example, (D x D)0
G1
in Lemma 1.1 with y satisfying the inequalities 0 < y < 1 or 1 < y < ro.
Let us go from C x C to XC by (1.3) and (1.4). The images of (D x D)0, (D' x D')0, (D x D')0, (D' x D)0 will ^e denoted by Y+ Y— ^- respectively.
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Bv (1.7) - (1.10) we get the Mowing description of these sets (recall that they all lie in XC : [x, x] = 1):
Y + : [x, x] > 1, Im — < 0,
x2
Y- : [x, x] > 1, Im — > 0,
x2
Q+ : —1 < [x, x] < 1, Imx1 > 0,
Q- : —1 < [x, x] < 1, Imx1 < 0,
The pairs (—iy, iy) and (—iy, iy-1) go to the points in XC lying on the curves
yt = (0, i sinh t, cosh t), ut = (i sin t, 0, cos t), (1.12)
where y = e-t, y = tan(n/4 — t/2), respectively. Representatives of the G1-orbits are the points:
yt : t> 0, for Y+,
yt : t< 0, for Y-,
ut : 0 < t < n/2, for Q+,
ut : —n/2 < t < 0, for Q-.
The Lie algebra 0 of the group G = SO0(1, 2) consists of matrices
( 0 ^1 6 \
X = I 6 0 —Co ) ■
VC2 Co 0/
Let us take the corresponding basis in 0:
0 0 0 0 1 0 0 0 1
L0 = I 0 0 —1 ) , L1 = I 10 0 ) , L2 = I 0 0 0 ) .
0 1 0 0 0 0 1 0 0
Let us give a remark about the manifolds Y± The Killing form of 0 is B(X, Y) = tr (XY), so that B(X, X) = 2(—Co + C2 + Cl) Consider in 0 two light cones (forward and backward) C+ and C- defined by the inequalities —C0 + Ci + C2 < 0, ±Co > 0.
These domains are invariant with respect to Ad G. In the complexification GC let
us take the two sets exp (iC±). It turns out that
Y± = X • exp (iC±).
Moreover, it turns out that the subsets r± = G • exp (iC±) of GC are semigroups (Olshanski).
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Now let us consider a complexifieation GC of the group G. It consists of complex
matrice of the third order preserving the form [x, y] in C3, Let us take the following
matrices in GC:
1 0 0
Yt = eitLo = I 0 cosh t —i sinh t ) ,
0 i sinh t cosh t
(cos t 0 i sin t \
0 10 ) . (1.13)
i sin t 0 cos t
xo = (0, 0, 1) E X
these matrices, i, e, yt = x°Yt, wt = x%.
Therefore, any point x in Y± is x0Ytg, where t > 0 for Y + and t < 0 for Y-, and
any point x in Q± is x05tg, where 0 < t < n/2 for Q+ and — n/2 < t < 0 for Q-,
gG
Let us return to the G^orbits of the pairs (0, ro) and (ro, 0) which were deleted from DxD' and D'xD respectively. Under the map (1.3) the pairs (0, ro) and (ro, 0) go respectively to the points un/2 = (i, 0, 0) = ix1 and w-n/2 = (—i, 0, 0) = —ix1, x1 = (1, 0, 0) G1 G
of the points ix1 and — ix1. Both points ±x1 belong to the hvperboloid [x, x] = — 1.
It consists of two the sheets L± so that x1 E L+ and —x1 E L-, Therefore the G-orbits are iL± They lie on the boundary of the manifolds Q± respectively. Each of them can be identified with the Lobachevsky plane L = G1/K1 = G/K.
All four complex manifolds (of real dimension 4) Y±, Q± are adjoint to the
X
manifolds Q^d Q- are adjoint to one sheet (to iL+ and iL-) of the two-sheeted [x, x] = —1
iL± the manifold Q± is a “complex crown” (after Akhiezer-Gindikin).
Let us assign to any point x in the manifolds Y±, Q± its third coordinate x3 E C.
Lemma 1.2 Under the 'map x M x3 the image of the manifold Y± is the whole
complex plane C with the cut [—1, 1], the image of the 'manifold Q± is the whole
complex plane with cuts (—ro, 1] and [1, ro).
Proof. For a point x E XC with coordinates z,w, see (1.3), we have
X3 + 1 = 1 + iz 1 — iw ^
x3 — 1 1 — iz 1 + iw
The function z M Z = (1 + iz)/(1 — iz) maps the disk D onto the right halfplane Re Z > 0, and to exterior D' onto the left half-plane Re Z < 0. Therefore, if
(z, w) E D x D (z, w) E D' x D'
(1.14) range over either the left or the right half-plane. Their product ranges over
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the whole plane C with cut (—ro, 0], If in addition z = w, then this product is not equal to 1, Hence x3 ranges over C without [—1, 1],
If (z, w) E D x D' or (z,w) E D' x D, then both fractions in the right-hand side of (1.14) range over different half-planes. Therefore, their product ranges over the whole of C with cut [0, ro), hence x3 ranges over C without (—ro, — 1] and [1, ro). Since we consider Q± but not D x D' and D' x D, we have to exclude in (1.14) pairs (z, w) for which w = 1/z. But it does not make the image smaller. Indead, if the first fraction in the right-hand side of (1.14) has value reia, — n/2 < a < n/2, then the second fraction has value —r-1eia, so that their product is equal to —e2i“. The intersection of the sets of these points over all a is empty, □
Let M(y) be a holomorphic function on the manifold Y± and let N(x) be its
X
M(x) = lim M(y), y E Y±, x E X.
y—x
We shall assume that y tends to x “along the radius”, i, e, if y E Y± and x E X have horospherieal coordinates (z,w) and (u,v) respectively, then
z = e-tu, w = e-tv (1,15)
and t M ±0. These equalities (1,15) give (for jt, see (1,13)):
y = xYt. (1-16)
Lemma 1.3 Let M(y) depend only on y3: M(y) = N(y3). By Lemma 1.2 the function N (A) is analytic on the plane C with c u,t [—1, 1]. Then one has
M(x) = N(x3 ^ i0x2).
yx
y3 = — i sinh t ■ x2 + cosh t ■ x3.
Therefore, y3 = —it ■ x2 + x3 + o(t), when t M 0, Hence the lemma. □
Now let M(u) be a holomorphic function on the manifold Q± and let M(x) be
X
M(x) = lim M(u).
y—^x
Here we assume similarly that u tends to x “along the radius”, i. e. if u E Q± and x E X have horospherieal coordinates (z,w) and (u,v) respectively, then
z = e-tu, w = etv (1-17)
t0
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Lemma 1.4 Let M(u) depend only on u3: M(u) = N(u3). By Lemma 1.2 the function N (A) is analytic on the plane C with cuts (—ro, — 1] and [1, ro). Then
M(x) = N(x3 ± i0 ■ x1x3) (1,18)
Proof, By (1.17) and (1,18) we have
1 + uv
u3
i(e-tu — etv)
u, v x
account the equality x1 + 1 = (x3 + ix2)(x3 — ix2), we obtain
u3 cosh t — i sinh t ■ x1. (1-19)
When t M 0, it behaves as x3(1 + itx1) up to terms of order t2. Hence the lemma. □
It is convenient to represent it using a cone in C4. Let us equip C4 with the bilinear form
[[x, y]] = —Xoyo — X1y1 + X2y2 + X3y3
(we add to vectors x in C3 the coordinate xo), Let C be the cone in C4 defined bv [[x, x]] = 0 x = 0, Then the complex hvperboloid XC is the section of the cone C by the hvperplane xo = 1. Looking at (1.3), consider the set Z of points
Z = 1 (i(z — w), z + w, i(1 — zw), 1 + zw), (1,20)
where z,w E C. It is the section of the cone C by the hyperplane —ix2 + x3 = 1, i, e, the hvperplane [x, £o] = 1, where £o = (0, 0,, — i, 1). The map Z M x = Z/Zo maps Z\{z = w} in XC, it gives just the horospherieal coordinates,
roZ
obtain D x D or D' x D', one has to the inequality [Z, Z] > 0 to add the inequality Im(Z3/Z2) < 0 or Im(Z3/Z2) > 0, respectively, and in order to obtain D x D' or D' x D, one has to the inequality [Z, Z] < 0 to add the condition that the imaginary part of the determinant
Co C1
Z o Z1
is less or greater than zero.
D x D
domain of type IV D(p) for p = 2. Indeed, D(p) is defined as follows: we equip Cn, n = p + 2, with the form [x, x] = —x2 — ... — x"^ + x2n-1 + x^. Let £o =
(0,..., 0, — i, 1). Then D(p) consists of points Z E Cn such that
[Z, Z] = 0, [Z, Z] > 0, [Z, £o] = 1, Im (Zn/Zn-1) < 0.
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Any point Z E D(p) can be written as
where Z1,... ,ZP satisfy the inequalities:
so that D(p) can be identified with a bounded domain in Cp,]
Let us assign to a point x = (xo,x1,x2,x3) in C4 the matrix
x
xo + ix1 x2 + ix3 x2 — ix3 xo — ix1
(1.21)
—[[x, x]] Z E Z
(1.20), one gets the matrix
These matrices Z are characterized by det Z = 0, Z12 = i-
The group G1 x G1 acts on the space of matrices (1.21): to an element (g1,g2) E G1 x G1 corresponds the linear transformation:
It is given by a real matrix of order 4. We obtain a homomorphism of the group G1 x G1 onto the group SOo(2, 2). The kernel is the group of order 2 consisting of the pairs (E, E), (—E, —E), The diagonal of G1 x G1; i, e. the set of pairs (g, g), g E G1; goes to the subgroup SOo(1, 2) = G, it preserves xo = (1/2) tr x.
G1 x G1 Z
Z
passing through the origin). This is the fractional linear action:
G1 x G1 M SOo(2, 2)
(we obtain a homomorphism SL(2, C) x SL(2, C) M SO(4, C)). Let us take in the eomplexifieation G^ x G^ the pairs
(1.22)
z M z ■ g2, w M w ■ g1.
(eitL°, eitL°) and (e“itL°, eitLo)
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Under the above homomorphism these pairs go to the matrices:
/ cosh t i sinh t 0 0 \
i sinh t cosh t 0 0 0 0 10.
\ 0 0 0 1 /
The first one is obtained bv bordering of the matrix Yt, see (1,13), The second one appears in the proof of Lemma 1,4, Indeed, let us multiply a vector (row) (1, x1, x2, x3) in C, such that the vector x = (x1, x2, x3) belongs to X, by this matrix and then divide by the coordinate with index zero, we just get a vector u E whose horospherical coordinates are connected with the horospherical coordinates x
u = — --- — (x1 ■ cosh t + i sinh t, x2, x3). (1,23)
cosh t — ix1 ■ i sinh t
It includes (1,19), The curve (1,23) with x = xo, i. e. the curve (i tanht, 0,1/cosh t), is in fact the curve ut, see (1.12), with another parameter.
§ 2. On the analytic continuation of spherical functions
First we consider spherical functions ^a,£(x), a = —1/2 + ip, e = 0,1, on the hyperboloid X of the continuous series. We study the analytic continuation of these functions to the complex manifolds defined in § 1,
In fact, we may consider the spherical functions ^a,£ on the hyperboloid X with generic a a E C, a E Z not only with a = —1/2 + ip.
These spherical functions ^a,£(x), a E C, e = 0,1, were computed in [4], [5], they are linear combinations of Legendre functions of the first kind of ±x3 (x3 being the
x
2n
^<T,E(x) =---:----- [Pa ( —x3) + (— 1)£Pa (x3)] . (2-1)
sin an
The Legendre function Pa(z) is analytic in the complex plane C with cut (—ro, —1]. At this cut, we define it as half the sum of the limit values form above and below:
Pa (c) = — [Pa (c + i0) + Pa (c — i0)] , C < —1.
Therefore, the combination Pa(—z) + (—1)£Pa(z) is analytic in C with cuts (—ro, —1] and [1, ro). Precisely these cuts appear in Lemmas 1,2 and 1,4 in treating the complex manifolds adjoint to X,
1 0 0 0 0 10 0
0 0 cosh t —i sinh t an
0 0 i sinh t cosh t
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So, if we want to continue the functions ^CT,£(x) analytically, then naturally should be taken for that purpose. Hence let us consider functions on defined by the same formula (2,1) with x replaced with u:
2n
^a,£(u) =------:----- [Pa (-u3) + (— 1)£Pa (u3)] . (2-2)
sin an
These functions are analytic on Let us determine their limit values at X,
First consider the function Pa (u3) on By Lemma 1.4 its limit values at X
are:
{Pa (X3), X3 > -1,
(2,3)
Pa (X3 T i0Xl), X3 < -1,
when u ^ X and u G It follows from [1] 3.3 (10) that the function Pa(z) has the following limit values at the cut (-ro, -1]:
Pa(c ± i0) = e±ianPa(-c) - — sin an ■ Qa(-c),
n
where c < -1 and Qa is the Legendre function of the second kind. Therefore, bv (2,3) we have for u ^ X, u G and X3 < -1:
lim Pa (шз)
2
Pa(—x3)-----sin anQa(—x3), xi > 0,
n
2
е±гажp^ (—x3)-----sin anQa (—x3), Xi < 0.
It can be written as
lim Pa(шз) = Pa(c) T sgnxi ■ isin an ■ Pa(-x3).
We see that the limit values of Pa (ш3) as ш ^ x coincide with Pa (x3) for x3 > — 1 only. In order to obtain Pa (x3) one has to take half the sum of the limit values from both manifolds Q+ and П-,
Similarly this goes for Pa(—y3).
Thus, the limit values at X of the function Ф^ (ш) on Q± defined by (2,2) coincide with the spherical function Ф^^) for —1 < x3 < 1 only. The spherical function ФayЄ(x) is even in xi; but the limit function lim Фа,є(ш) does not. In order to obtain the spherical function Фа(x) from the function Фaє(ш), one has to use both manifolds Q± and to take half the sum of the limit values from П+ and П-:
Ф^є^) = 1 ^ lim Фа,є(ш),
2
where the limit is taken when ш ^ x, ш G Q±, x gX.
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Now we consider the spherical functions ^n,± of the holomorphic and anti-holomorphic discrete series on X. Here n G N = {0,1, 2,...}. The signs “+" and “-" correspond to the holomorphic and the anti-holomorphic series respectively. These functions are expressed [4] in terms of Legendre functions of the second kind:
^™,±(x) = 4Qra(X3 T i0 ■ X2).
Comparing it with Lemma 1,3, we see that the spherical function ^n,± is continued analytically on the manifold Y± as the function
^n(y) = 4Qn(y3).
References
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3, V, F, Molchanov, Holomorphic discrete series for hyperboloids of Hermitian type, J, Funct, Anal,, 1997, vol. 147, No, 1, 26-50
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